Ive always found that bullies are that way partly because they've made a habit out of it, and get somewhat stuck in bullying. Its what they are used to and what people expect from them, so people generally aren't too nice, or willing to open up and talk to them either.
What I do is go out of my way to be nice to them. Sure, they might try to make fun of you for doing so, but you just go about your day. Do it a few times, and these people will have no choice but to reciprocate the action by being nice in turn.
The key to bullies in general tho really is:Do not react when they try to be an a-hole to you. They always do it for the reaction, and they choose the people they do because of the way they react to the bullying. Unfortunately, the people who don't react well usually don't have great social skills to start, so it perpetuates the bullying...
Overall though, its no fun to bully someone after they just did something nice for you. They might be able to do it a few times, but once a personal relationship is developed just by you doing a few nice things, it will dissipate. I promise that this technique works. Im in sales, and sales departments are very common spots for bullies to end up, so i have helped a LOT of people deal with workplace bullying. Its gonna feel weird at first doing nice things for them (I promise that too) BUT things will never change if you don't change your approach, and there are people like this everywhere so running wont work either.
I hope someone sees this who this will help, because i like to have fun and all that but i HATE seeing bullying honestly... It just always seems so one sided and unfair. Like you might have good social skills mr bully, but how well do you adult??? Do you pay your bills? Balance your checkbook? Do you know how to code? The people who get bullied just lack in social skills a lot of the time but they are probably way more advanced in other areas...